## Abstract

In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs. Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding f: G(V, E) T has Ramsey hop-distortion (t, M,β, h), (here t, β, h≥q 1 and M subseteq V) if forall u\in M, v\in V,\dG(β h)}(u, v)≤q dT(u, v)≤q t dG(h)(u, v). t is called the distortion, β is called the hop-stretch, and dG(h)(u, v) denotes the minimum weight of a u-v path with at most h hops. Haeupler et al. constructed embedding where M contains 1-ϵ fraction of the vertices and β=t=O(log2nϵ). They used their embedding to obtain multiple bicriteria approximation algorithms for hop-constrained network design problems. In this paper, we first improve the Ramsey-type embedding to obtain parameters t=β= frac tildeO(n) ϵ, and generalize it to arbitrary distortion parameter t (in the cost of reducing the size of M). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler et al. Further, we construct hop-constrained clan embeddings (where each vertex has multiple copies), and use them to construct bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the non constrained version. Finally, we use our embedding results to construct hop constrained distance oracles, distance labeling, and most prominently, the first hop constrained compact routing scheme with provable guarantees. All our metric data structures almost match the state of the art parameters of the non-constrained versions.

Original language | English |
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Title of host publication | Proceedings - 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science, FOCS 2021 |

Publisher | IEEE Computer Society |

Pages | 492-503 |

Number of pages | 12 |

ISBN (Electronic) | 9781665420556 |

DOIs | |

State | Published - 2022 |

Event | 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 - Virtual, Online, United States Duration: 7 Feb 2022 → 10 Feb 2022 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2022-February |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 |
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Country/Territory | United States |

City | Virtual, Online |

Period | 7/02/22 → 10/02/22 |

### Bibliographical note

Publisher Copyright:© 2022 IEEE.

## Keywords

- Approximation algorithms
- Compact routing scheme
- Distance labelings
- Distance oracle
- Group Steiner tree
- Hop constrained tree embeddings
- Metric embeddings